Power functions with negative exponents

Functions: Fractional functions

Power functions with negative exponents

Power function with negative exponent

A power function with a negative integer exponent has the form \[f(x)=\blue{a}x^{-\orange{n}}\]

in which #\orange{n}# is a postive integer.

We can also write this function as \[f(x)=\frac{\blue{a}}{x^{\orange{n}}}\]

The graph of a power function with a negative integer exponent, moves through the point #\rv{1,\blue{a}}#, has a vertical asymptote at #x=0# and horizontal asymptote in the line #y=0#.

If #\orange{n}# is even, the function is symmetrical across the #y#-axis. If #\orange{n}# is odd, the function has the point #\rv{0,0}# as the point of symmetry.

GeoGebra Negative powerfunction

Do‍main and rangeA power function with negative exponent has a vertical asymptote at #x=0#. This is so because we are not allowed to divide by #0#, hence, the function value at #f(0)# does not exist. The domain of a power function with negative exponent are all numbers, except #x=0#.

The range of a power function with an even negative exponent is equal to all positive numbers if #\blue a \gt 0#. If #\blue a \lt 0#, then the range is equal to all negative numbers.

For a power function with odd negative exponent, the range is equal to all numbers except #0#.

Ev‍en and odd function

When #\orange{n}# is even, we call #f(x)=\frac{\blue{a}}{x^{\orange{n}}}# an even function. This means that: \[f(-x)=f(x)\]

Hence, this is another way of describing that the graph is symmetrical across the #y#-axis.

When #\orange{n}# is odd, we call #g(x)=\frac{\blue{a}}{x^{\orange{n}}}# an odd function. This means that: \[g(-x)=-g(x)\]

Hence, this is another way of describing that the graph has point symmetry across the point #\rv{0,0}#.

Example

For #f(x)=\frac{\blue{1}}{x^{\orange{4}}}# we have:

\[f(-2)=\frac{\blue1}{\left(-2\right)^{\orange{4}}}=\frac{1}{16}=\frac{\blue1}{\left(2\right)^{\orange{4}}}=f(2)\]

For #g(x)=\frac{\blue1}{x^{\orange{3}}}# we have:

\[f(-2)=\frac{\blue1}{\left(-2\right)^{\orange{3}}}=\frac{1}{8}=-\frac{\blue1}{\left(2\right)^{\orange{3}}}=-f(2)\]

Take a look at the graph of a power function with negative exponent, which is a function of the form #f(x)=\frac{a}{x^n}#.

What do we know about the values of #n# and #a#?

The value of #n# is: odd
The value of #a# is: positive

The graph is symmetrical across the point #\rv{0,0}#, hence, the value of #n# is odd.
The #y#-value is positive if the value of #x# is positive, hence the value of #a# is positive.

New example